Arithmetic Practice

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Difficulty

Direct

1.1Write 8,405.72 in Expanded Form

Exam I | Problem 1.1 |

Write $8{,}405.72$ in expanded form.

Solution

Break the number into the value of each digit:

$$ 8{,}405.72 = 8000 + 400 + 5 + 0.7 + 0.02 $$

This matches the thousands, hundreds, ones, tenths, and hundredths places.

1.2Compare Two Integers

Exam I | Problem 1.2 |

Fill in the correct symbol:

-3\ \_\ -8

Solution

Compare the numbers on a number line. Since $-3$ is to the right of $-8$, it is greater.

$$ -3 > -8 $$

1.3Find an Absolute Value

Exam I | Problem 1.3 |

Absolute Value

Evaluate:

$$ |-14| $$

Solution

Absolute value is distance from zero, so it is always nonnegative.

$$ |-14| = 14 $$

1.4Add Whole Numbers

Exam I | Problem 1.4 |

Compute:

$$ 467 + 289 $$

Solution

Add by place value:

$$ 7 + 9 = 16 $$

Write $6$ and carry $1$.

$$ 6 + 8 + 1 = 15 $$

Write $5$ and carry $1$.

$$ 4 + 2 + 1 = 7 $$

So,

$$ 467 + 289 = 756 $$

1.5Add Decimals

Exam I | Problem 1.5 |

Compute:

$$ 13.6 + 4.75 $$

Solution

Line up the decimal points:

$$ 13.60 + 4.75 $$

Add:

$$ 13.60 + 4.75 = 18.35 $$

1.6Subtract Whole Numbers

Exam I | Problem 1.6 |

Compute:

$$ 700 - 458 $$

Solution

Regroup because $0 < 8$.

700 = 6\text{ hundreds} + 9\text{ tens} + 10\text{ ones}

Now subtract by place value:

$$ 10 - 8 = 2 $$

$$ 9 - 5 = 4 $$

$$ 6 - 4 = 2 $$

So,

$$ 700 - 458 = 242 $$

1.7Multiply by a One-Digit Number

Exam I | Problem 1.7 |

Compute:

8 \times 7

Solution

Multiply directly or think of repeated addition:

8 \times 7 = 8 + 8 + 8 + 8 + 8 + 8 + 8 = 56

1.8Divide Evenly

Exam I | Problem 1.8 |

Compute:

42 \div 7

Solution

Find the number that times $7$ gives $42$:

7 \times 6 = 42

So,

42 \div 7 = 6

1.9Evaluate a Basic Expression

Exam I | Problem 1.9 |

Order of Operations

Compute:

$$ 3(4+5)-2 $$

Solution

Follow the order of operations.

First evaluate the parentheses:

$$ 4+5=9 $$

Then multiply:

$$ 3(9)=27 $$

Finally subtract:

$$ 27-2=25 $$

1.10Decide Whether 29 Is Prime or Composite

Exam I | Problem 1.10 |

Prime and Composite Numbers

Is $29$ prime or composite?

Solution

Check whether $29$ has any factors besides $1$ and itself. It is not divisible by $2$, $3$, or $5$, so its only positive factors are $1$ and $29$.

Therefore, $29$ is prime.

Difficulty

Integrated

2.1Use Signed Addition and Subtraction

Exam II | Problem 2.1 |

Compute:

$$ -6 + 11 - 4 $$

Solution

First add the first two numbers:

$$ -6 + 11 = 5 $$

Then subtract $4$:

$$ 5 - 4 = 1 $$

So the value is $1$.

2.2Find the Greatest Common Factor

Exam II | Problem 2.2 |

Find the greatest common factor of $18$ and $24$.

Solution

List the factors:

$$ 18: 1, 2, 3, 6, 9, 18 $$

$$ 24: 1, 2, 3, 4, 6, 8, 12, 24 $$

The largest factor they share is $6$.

So,

$$ GCF(18,24)=6 $$

2.3Convert a Fraction to a Decimal and Add

Exam II | Problem 2.3 |

Compute:

0.2 + \frac{3}{8}

Solution

Convert the fraction to a decimal:

\frac{3}{8} = 0.375

Now add:

$$ 0.2 + 0.375 = 0.575 $$

2.4Multiply Fractions

Exam II | Problem 2.4 |

Compute:

\frac{2}{3}\cdot\frac{9}{10}

Solution

Multiply numerators and denominators:

\frac{2}{3}\cdot\frac{9}{10}=\frac{18}{30}

Simplify:

\frac{18}{30}=\frac{3}{5}

2.5Find a Unit Rate

Exam II | Problem 2.5 |

A car travels $180$ miles in $3$ hours. What is its unit rate in miles per hour?

Solution

Divide miles by hours:

180 \div 3 = 60

So the unit rate is $60$ miles per hour.

2.6Solve a Proportion

Exam II | Problem 2.6 |

Solve for $x$:

\frac{x}{15}=\frac{4}{5}

Solution

Cross multiply:

5x = 15 \cdot 4

$$ 5x = 60 $$

Divide by $5$:

$$ x = 12 $$

2.7Estimate and Then Multiply

Exam II | Problem 2.7 |

Estimate $49 \times 21$ using compatible numbers, then find the exact product.

Solution

Round to nearby easy numbers:

49 \approx 50,\quad 21 \approx 20

Estimate:

50 \times 20 = 1000

Now compute exactly:

49 \times 21 = 49(20+1)=980+49=1029

The estimate is close to the exact value.

2.8Use Square and Cube Roots

Exam II | Problem 2.8 |

Compute:

\sqrt{49}+\sqrt[3]{27}

Solution

Evaluate each root:

\sqrt{49}=7

and

\sqrt[3]{27}=3

Add the results:

$$ 7+3=10 $$

Difficulty

Applied

3.1Apply a Scale Factor

Final | Problem 3.1 |

A drawing is enlarged by a scale factor of $3$.

If a segment is $4$ cm long in the original drawing, how long is it in the enlarged drawing?

Solution

Multiply the original length by the scale factor:

3 \cdot 4 = 12

So the new length is $12$ cm.

3.2Compute Simple Interest

Final | Problem 3.2 |

Find the simple interest on $\$450$ at $6\%$ for $2$ years.

Solution

Use the formula $I=Prt$:

$$ I = 450(0.06)(2) $$

$$ I = 54 $$

The interest is $\$54$. The total amount is

$$ 450 + 54 = 504 $$

So the total is $\$504$.

3.3Find a Percent Increase

Final | Problem 3.3 |

A jacket costs $\$80$ and is marked up by $15\%$.

What is the new price?

Solution

Find $15\%$ of $80$:

$$ 0.15(80)=12 $$

Add the increase to the original price:

$$ 80+12=92 $$

The new price is $\$92$.

3.4Write a Large Number in Standard Form

Final | Problem 3.4 |

Scientific Notation

Write $7.4 \times 10^5$ in standard form.

Solution

Multiply $7.4$ by $100{,}000$:

7.4 \times 10^5 = 740{,}000

3.5Use Division with a Remainder

Final | Problem 3.5 |

A school buys $95$ pencils in packs of $8$.

How many full packs can it make, and how many pencils remain?

Solution

Divide to find the number of full packs:

95 \div 8 = 11\text{ remainder }7

This means there are $11$ full packs and $7$ pencils left over.

Difficulty

Challenge

4.1Mix Exponents, Roots, and Order of Operations

Final | Problem 4.1 |

Compute:

2^3 + 3(5-2)^2 - \sqrt{81}

Solution

Follow the order of operations.

First evaluate the parentheses:

$$ 5-2=3 $$

Then the exponent:

$$ 3^2=9 $$

Now multiply:

$$ 3(9)=27 $$

Evaluate the other exponent and root:

$$ 2^3=8 $$

\sqrt{81}=9

Finish by adding and subtracting:

$$ 8+27-9=26 $$

4.2Apply Two Percent Changes in a Row

Final | Problem 4.2 |

A $\$50$ item is increased by $20\%$ and then decreased by $10\%$.

What is the final price?

Solution

First find the $20\%$ increase:

$$ 0.20(50)=10 $$

So the new price is

$$ 50+10=60 $$

Now find the $10\%$ decrease of $60$:

$$ 0.10(60)=6 $$

Subtract:

$$ 60-6=54 $$

The final price is $\$54$.

4.3Combine Fractions and Decimals

Final | Problem 4.3 |

Compute:

\frac{3}{4} + 0.6 - \frac{5}{8}

Solution

Convert the fractions to decimals:

\frac{3}{4}=0.75

and

\frac{5}{8}=0.625

Now calculate:

$$ 0.75 + 0.6 - 0.625 $$

$$ = 1.35 - 0.625 $$

$$ = 0.725 $$

4.4Estimate and Compute a Scientific-Notation Product

Final | Problem 4.4 |

Estimate and then compute:

(4.5 \times 10^3)(2 \times 10^2)

Solution

Estimate by rounding $4.5$ to $5$:

(5 \times 10^3)(2 \times 10^2) = 10 \times 10^5 = 10^6

Now compute exactly:

(4.5 \times 2)\times 10^{3+2} = 9 \times 10^5

So the exact product is

9 \times 10^5 = 900{,}000